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arX
iv:1
201.
1730
v3 [
hep-
ph]
15
Oct
201
2SCIPP 12/01, UCI-TR-2012-01
revised: October, 2012
Group-theoretic Condition for Spontaneous CP Violation
Howard E. Haber1 and Ze’ev Surujon2
1Santa Cruz Institute for Particle Physics,
University of California, Santa Cruz, CA 95064, USA
2Department of Physics and Astronomy,
University of California, Irvine, CA 92697, USA
Abstract
We formulate the necessary conditions for a scalar potential to exhibit spontaneous CP violation.
Associated with each complex scalar field is a U(1) symmetry that may be explicitly broken by
terms in the scalar potential (called spurions). In order for CP-odd phases in the vacuum to
be physical, these phases must be related to spontaneously broken U(1) generators that are also
explicitly broken by a sufficient number of inequivalent spurions. In the case where the vacuum
is characterized by a single complex phase, our result implies that the phase must be associated
with a U(1) generator that is broken explicitly by at least two inequivalent spurions. A suitable
generalization of this result to the case of multiple complex phases has also been obtained. These
conditions may be used both to distinguish models capable of spontaneous CP violation, and as a
model building technique for obtaining spontaneously CP-violating deformations of CP conserving
models. As an example, we analyze the generic two Higgs doublet model, where we also carry out
a complete spurion analysis. We also comment on other models with spontaneous CP violation,
including the chiral Lagrangian, a minimal version of Nelson-Barr model, and little Higgs models
with spontaneous CP violation.
1
I. INTRODUCTION
In the Standard Model (SM), CP invariance is broken explicitly by the Cabibbo-
Kobayashi-Maskawa (CKM) phase. Models beyond the SM often introduce additional
CP-odd phases. For example, these new sources of CP violation are needed to explain the
baryon asymmetry of the universe [1]. However, the observation of CP-violating phenomena
does not necessarily imply that the fundamental source of CP non-invariance is due to the
explicit breaking of CP. In particular, CP-violating phenomena may be a consequence of
spontaneous CP violation, where the Lagrangian of the theory respects the CP symmetry
but the vacuum is not invariant under CP. Such a case arises when the vacuum expectation
value (VEV) of a scalar field operator exhibits physical CP-odd phases (which cannot be
removed from the theory by field re-definitions). Models in which all CP-odd phases, in-
cluding the CKM phase, are due to spontaneous CP violation, have the potential of solving
the strong CP problem, as exhibited by the Nelson-Barr models [2].
Explicit CP violation may be established by proving the non-existence of a real basis, i.e.,
a basis in field space where all couplings are real. A basis-independent approach is one that
identifies basis invariant quantities that are CP odd. Perhaps the best-known example is the
Jarlskog invariant [3] of the SM. In contrast, the case of spontaneous CP violation is more
complicated. In its simplest form, spontaneous CP violation (SCPV) occurs if and only if
a class of real bases exists (which implies that the Lagrangian respects the CP symmetry),
but no real basis exists in which all the VEVs are simultaneously real-valued. In this case,
the CP-odd invariants depend on the Lagrangian couplings both explicitly and implicitly
via the VEVs [4, 5]. Therefore, they are generally complicated functions of the model
parameters. Moreover, it is difficult to systematize the construction of these invariants in a
model-independent way.
It is the purpose of this paper to provide a model-independent formulation of the necessary
conditions for spontaneous CP violation. These conditions derive from the fact that any
phase in the VEV must be related to a spontaneously broken U(1) generator. In order for
this phase to be physical, it is clear that the associated U(1) symmetry should be broken
explicitly. The coefficients of the corresponding U(1) breaking terms that appear in the
Lagrangian will henceforth be called spurions, since the explicit U(1) symmetry breaking
can be formally restored by assigning appropriate transformation laws to these coefficients
(in particular, spurions by definition carry a nonzero U(1) charge). However, in order to
guarantee that the phase in the VEV is physical, there must be a sufficient number of
inequivalent spurions relative to the number of broken U(1) generators. For example, a
single complex VEV can give rise to SCPV only if the associated U(1) is broken explicitly
by at least two spurions whose U(1) charges differ in magnitude. In theories of multiple
complex scalars, there is a different U(1) associated with each complex field. Each spurion
is characterized by a charge vector, whose components are the corresponding U(1) charges.
SCPV can arise only if the number of spurions Ns is larger than the maximal number
2
of linearly independent charge vectors, denoted by r. Moreover, the number of potential
CP-violating phases is determined to be equal to r − r′, where r′ is the number of charge
vectors that are linearly independent of the remaining Ns − 1 charge vectors. A geometrical
interpretation of this result is provided in Appendix A.
The practical implication of this formulation is three-fold. First, it provides a simple
way to find out whether a potential CP-odd phase in the VEV is physical generically as
a function of the model parameters. Second, it provides a clearer way to understand why
certain regions in the parameter space never exhibit spontaneous CP violation while others
may do so. Finally, given a CP-conserving model, our condition can be used to find a
deformation of that model which is spontaneously CP-violating in a generic region of its
parameter space.
In this paper, we first compare and contrast explicit and spontaneous CP violation in
Section II. In Section III, we discuss in detail the necessary conditions for spontaneous CP
violation. We illustrate these conditions in Section IV by applying our results to the two
Higgs doublet model (2HDM) [4–9]. In this analysis, the relevant explicitly-broken U(1)
symmetry is the Peccei-Quinn symmetry [10, 11]. In Section V, we exhibit our conditions
in other models of spontaneous CP-breaking, by considering the chiral Lagrangian [12],
the minimal Nelson-Barr model [13], and the spontaneously CP violating Littlest Higgs [14].
Applying our formulation provides new insight to the question of spontaneous CP violation in
these models. Our conclusions and future directions are given in Section VI. In Appendix B,
we exhibit the full power of the spurion analysis for the 2HDM, in which we examine spurions
with respect the full SU(2) Higgs flavor group. We reproduce results previously obtained
by Ivanov [15] and show how this formalism can be used for constructing basis-independent
invariants. We also provide a more transparent understanding of the basis-independent
condition for the existence of the U(1) Peccei-Quinn symmetry in the 2HDM.
II. EXPLICIT AND SPONTANEOUS CP VIOLATION
The question of whether CP is violated explicitly or spontaneously deserves some care
due to the basis-dependence associated with the definition of CP. For simplicity, we focus
in this section on scalar field theories, with scalar fields φi(~x, t), for i = 1, 2, . . . n. Consider
the following generalized CP-transformation (GCP) [16–21],
φi(~x, t) −→ Xijφ∗j (−~x, t) . (1)
where X is an n× n unitary matrix. Such a transformation is automatically a symmetry of
the free scalar field theory action. The form of this generalized CP transformation is basis-
dependent. Namely, one can redefine the scalar fields such that φ′i(x) = Uijφj(x), where U
is an arbitrary n×n unitary matrix. The GCP-transformation in terms of the primed fields
3
is of the form given by eq. (1), where X is replaced by
X ′ = UXUT . (2)
The interacting scalar field theory is GCP-invariant if the action is invariant under eq. (1) for
some choice ofX . Three classes of GCP transformations exist: (i)XX∗ = 1; (ii)XX∗ = −1;and (iii) XX∗ 6= ±1 (denoted in [20, 21] as CP1, CP2 and CP3, respectively), where 1 is
the n × n identity matrix. However, any CP2 or CP3 scalar field theory also respects CP1
(henceforth denoted as CP). Hence, in what follows we focus on the case where XX∗ = 1,
which implies that X is a symmetric unitary matrix. We now employ the well known result
that any symmetric unitary matrix X can be written as the product of a unitary matrix
and its transpose (see e.g. Appendix D.3 of [22] for a proof of this result). That is, one can
always find a unitary matrix U such that X = U †U∗. Using eq. (2), it then follows that:
X ′ ≡ UXUT = UU †(UU †)∗ = 1 . (3)
That is, for any CP-invariant scalar field theory, there is always a basis choice for which
X ′ = 1, in which case the CP transformation reduces to complex conjugation and inversion
of the space coordinate.
1. Explicit CP Violation (XCPV): If a basis transformation U can be found such
that the scalar field theory action is invariant under eq. (1) with X = 1, then there exists
a real basis, i.e., a basis where all the couplings are real and the model is explicitly CP
conserving. Conversely, if eq. (1) is not a symmetry of the scalar field theory action for any
choice of the unitary matrix X , then no real basis exists and the scalar field theory explicitly
violates CP.
If the scalar field theory model is explicitly CP conserving, then a real basis exists,
with corresponding GCP transformation X = 1. Consider the set of basis transformations
denoted by Ur that maintain the real basis. This set necessarily includes all real orthogonal
n × n matrices. Applying eq. (2), we see that X = 1 in a real basis related to the original
one by a real orthogonal basis change. Depending on the form of the interacting scalar
Lagrangian, the set Ur may also include a subset of the unitary n × n matrices, denoted
by Us, that are not real (and hence are not orthogonal). In this case, the corresponding
X 6= 1. Consider the ground state of a scalar field theory determined by a set of VEVs,
〈φi〉 ≡ vi. If the vacuum is GCP-invariant, then vi = Xijv∗j .
2. Spontaneous CP Violation (SCPV): Given an explicitly CP-conserving scalar
field theory, the vacuum is CP-invariant if and only if a real basis exists in which all the
scalar field VEVs are real (cf. Theorem 3 in Appendix F of [5]). Suppose that a real basis is
chosen such that X = 1 and the scalar field VEVs are not all real. It still may be possible
to find a set of the basis transformations Us that preserve the real basis such that all
the scalar field VEVs are real. In this case, the scalar field theory and the vacuum are
CP-conserving. If the set Us is empty, then the model is said to exhibit spontaneous CP
violation (SCPV).
4
Note that if the model is explicitly CP violating (i.e., there is no real basis: the set Ur is
empty), then the question of spontaneous CP violation is no longer meaningful, since there
is no well-defined CP transformation law that one can apply to the vacuum.
III. NECESSARY CONDITIONS FOR SPONTANEOUS CP VIOLATION
Given a real basis, spontaneous CP Violation is triggered by physical phases in the VEVs.
We shall now examine what this implies for global symmetries and their breaking.
A. A Single Complex Scalar
Consider a complex scalar field degree of freedom, φ, and the associated field redefinition
φ → eiαφ. This set of possible field redefinitions is a U(1) subgroup of the maximal global
symmetry group O(2) of the kinetic energy terms. Define the generator X of this field
redefinition, such that φ is charged under U(1)X while the other degrees of freedom are
neutral. For certain potentials, the field φ acquires a VEV, 〈φ〉 = veiθ, breaking U(1)Xspontaneously. In this case, it is useful to parameterize the field in angular variables,
φ(x) = ρ(x)eiG(x)/v , (4)
where G is a periodic field, G ∼ G + 2πv. As long as U(1)X is not broken explicitly, G is
an exact Goldstone boson. It shifts under U(1)X according to G → G+ vα. This induces a
shift in the phase of the VEV, θ → θ + α, which defines a circle of equivalent vacua. Any
phase θ0 is then unphysical, since it is equivalent to θ = 0 by a U(1)X transformation which
is an exact symmetry.
There are two possible ways to remove the Goldstone mode G from the spectrum. First,
one may gauge U(1)X , so that G becomes the longitudinal component of the associated
gauge boson. Note that in this case, the phase is still unphysical: it can be removed by a
gauge transformation. A second possibility is to introduce explicit breaking of U(1)X , such
that G becomes a massive pseudo-Goldstone boson.
An explicit breaking of U(1)X introduces a potential for the otherwise flat Goldstone
direction in field space. Then one may ask whether G acquires a VEV with a non-zero
physical phase. As a first attempt, suppose that U(1)X is broken by a single term in the
potential,
VX = 12bφ2 + h.c., (5)
where b is real valued. Apart from this term, the Lagrangian depends only on ∂µG. The
new term introduces the only non-derivative dependence on G,
VX = bρ2 cos2G
v, (6)
5
and is minimized at θ = 〈G〉/v = π/2. However, the phase can be removed by the field
redefinition G → G− vπ/2, which is equivalent to φ → −iφ. This transformation induces a
sign flip, b → −b, such that in the new basis, the minimum is at θ = 0 and there is no CP
violation.
Had we introduced a different (higher) power of the field, e.g., gφ4, there would still be
a field redefinition (in this case: φ → e−iπ/4φ) which removes the phase from the VEV.
Note that while this transformation is not a symmetry, it leaves the Lagrangian parameters
real, merely changing the sign of g, while removing the phase from the VEVs. This is true
for any single monomial gkφk. The reason for this is that such a term always gives rise
to a pure cosine potential, Vk(θ) = 2gkvk cos(kθ). Since this potential has the property
Vk(θ + π/k) = −Vk(θ), one can always choose a basis where the minimum is at the origin,
which implies that the vacuum conserves CP.
If we introduce two terms with different powers of φ, the resulting potential for θ becomes
a more general function, whose minimum cannot generically be shifted to the origin without
introducing a phase difference among the couplings. Here the word “generically” should be
interpreted as: “in an O(1) fraction of the parameter space”. As an example, consider
VX = bφ2 + gφ4 + h.c., (7)
where b and g are real. The new terms induce a potential for the otherwise flat θ, which is
given by
VX = bv2 cos(2θ) + gv4 cos(4θ). (8)
For parameters in the range |b| < 4gv2, this potential is minimized at
cos(2θmin) = −b/(4gv2), (9)
generically resulting in spontaneous CP violation.
Although VX given in eq. (7) provides an explicit violation of the U(1)X global symmetry,
we can formally make VX neutral under U(1)X by assigning two different U(1)X charges to
the coefficients b and g. Indeed, if b → e−2iαb and g → e−4iαg, then VX is formally invariant
under the U(1)X transformation φ → eiαφ. One can interpret b and g as vacuum expectation
values of two new scalar fields Φb and Φg respectively, in which case the explicit breaking of
U(1)X is reinterpreted as the spontaneous breaking of U(1)X due to the nonzero VEVs for
the fields Φb and Φg. In the literature, the VEVs b ≡ 〈Φb〉 and g ≡ 〈Φg〉 are commonly called
spurions. Thus, in the above example, the spontaneous breaking of CP is attributed to the
breaking of the U(1)X symmetry by two spurions whose U(1)X charges differ in magnitude.
Note that for any spurion with U(1)X charge q, there is a complex conjugated spurion with
U(1)X charge −q. Hence, it is the magnitude of the charge that is relevant for determining
whether SCPV is possible. Thus, we arrive at the following necessary condition for SCPV
in the case of a single complex scalar field:
6
Spontaneous CP violation in a theory of single complex scalar field may occur only if the
related U(1) is broken by at least two spurions whose U(1) charges differ in magnitude.
Note that the value of the CP-violating phase in eq. (9) does not vanish in the b, g → 0
limit, as long as b/(gv2) ∼ 1. This may seem strange at first sight, but it can be understood
as follows. Without the explicit breaking, the phase is not physical and can take on any
value. With explicit breaking present, no matter how small, the phase becomes physical
and its value is stabilized by the effective potential. That is, the explicit breaking terms
break the degeneracy of the unperturbed problem (in which the energy is independent of
the phase θ). This is typical of all degenerate perturbation theory problems in quantum
mechanics. Indeed, one can see that for b/v2, g ≪ 1 with b/(gv2) ∼ 1, the depth of the
θ-dependent part of the potential, eq. (8), is of order gv4 ∼ bv2. Thus, the CP-violating
phase becomes meaningless in any physical process whose characteristic energy (or mass) is
larger than g1/4v.
B. Multiple Complex Scalars
In a model with multiple complex scalar fields, the vacuum may be characterized by more
than one CP-violating phase. Although the value of any specific phase is basis-dependent,
the number of potential1 CP-violating phases is well-defined and basis-independent.
The analysis of Section III.A shows that in a model with a single complex scalar field,
the spurions are labeled by their U(1) charge and SCPV requires at least two spurions with
U(1) charges of different magnitude. If this latter condition is satisfied, then the vacuum
is characterized by at most one independent CP-odd phase. In the case of N complex
scalar fields, the maximal symmetry group of the kinetic energy terms is O(2N), whereas
the number of independent physical phases cannot exceed N . These phases can always be
taken to be the “diagonal” phases associated with the Cartan subgroup U(1)1×· · ·×U(1)N ,
where each U(1) rotates the phase of one complex degree of freedom.2 If the scalar potential
contains Ns inequivalent spurions, then each spurion may be labeled by an N -dimensional
charge vector whose jth component is the charge under the U(1)j. Two spurions will be
considered to be “equivalent” if their charge vectors are equal up to a possible overall minus
sign.3
We construct the Ns × N matrix whose rows are given by the charge vectors of the
spurions. The rank r of this matrix is equal to the dimension of the vector space spanned
by the corresponding charge vectors. Since the rank of a matrix cannot exceed the number
1 To determine whether a potential phase is physical, one must minimize the effective potential of the phases
to check for nontrivial solutions.2 Here we assume that none of the N generators are gauged. If some of them are, the relevant group would
be smaller.3 As previously noted, the charge vector of a complex conjugated spurion is equal to the negative of the
charge vector of a spurion. Thus, we consider a spurion and its charge conjugate to be equivalent in the
present analysis. 7
of columns or rows, it follows that r ≤ min Ns, N. The physical interpretation of the
rank is easily discerned. Namely, only r independent U(1)’s are broken by the spurions,
which leaves N − r unbroken U(1)’s. Hence, one can define new U(1) generators that are
linear combinations of the original U(1) generators such that the first r U(1) generators are
explicitly broken and the last N − r U(1) generators are unbroken. In particular, the last
N − r components of the charge vectors of the spurions with respect to the new set of U(1)
generators are zero.
Thus, without loss of generality, one can simply consider truncated r-dimensional charge
vectors (where the last N − r zeros are removed). Indeed, there can be at most r physical
CP-violating phases associated with the N complex scalar degrees of freedom, since N − r
phases can be removed by employing the unbroken U(1)’s. We shall denote the truncated
r-dimensional charge vectors by
q(i) ≡(q(i)1 , q
(i)2 , . . . , q(i)r
), i = 1, . . . , Ns. (10)
As above, we can assemble the truncated charge vectors into an Ns × r matrix whose ith
row is given by q(i), which we denote by Q. By construction, r = rkQ and Ns ≥ r.
Consider first the case where Ns = r. This means that the Ns vectors q(i) are linearly
independent and therefore Q is an invertible r × r matrix. It is convenient to redefine the
U(1)r generators X1, . . . , Xr by X ′i ≡
∑j CijXj , where C = (QT)−1. Relative to this new
basis for the U(1) generators, the charge vectors are given by
δij =r∑
k=1
Cjkq(i)k , i, j = 1, . . . , r. (11)
Consequently, we have reduced the problem to r independent copies of one complex
scalar field and associated spurion (and its complex conjugate). In particular, if we denote
〈φn〉 = vneiθn, then the multi-field generalization of eq. (6) is given by
VX′
1,X′
2,...,X′
r=
r∑
i=1
Vi(vn) cos θ′i , where θ′i ≡
r∑
k=1
q(i)k θk , (12)
where Vi(vn) is the contribution to the potential of the ith spurion (where the complex fields
φn are replaced by the vn, respectively). Using the results of Section III.A, we conclude that
no physical phases exist in the vacuum and thus there is no SCPV.
In the case of Ns > r, we first label the truncated r-dimensional charge vectors such that
q(1), q(2), . . . , q(r) are linearly independent. Then, the charge vectors of the remaining
spurions, q(i) for i = r+1, r+2, . . . , Ns, are linear combinations of the first r charge vectors.
This means that that if we only keep the (inequivalent) spurions labeled by i = 1, 2, . . . , r,
we would again conclude that no physical phases exist in the vacuum. Hence, if we include
all Ns inequivalent spurions, we are left with at least one potential physical phase. To
determine whether SCPV actually occurs, one must minimize the effective potential as in
8
the single complex field case to determine the vacuum value of this phase. We conclude the
following:
SCPV may occur only if the number of inequivalent spurions is larger than the dimension
of the vector space spanned by the corresponding charge vectors.
In a model of multiple complex scalar fields with Ns > r, the number of potential physical
phases (henceforth denoted by d) is obtained as follows. In analogy with eq. (11), we define
new charge vectors with respect to the redefined U(1) generators X1, . . . , Xr,
r∑
k=1
Cjkq(i)k =
δij , for i = 1, 2, . . . , r ,
q′ (i)j , for i = r + 1, r + 2, . . . , Ns ,
(13)
where C = (QT)−1 and Q is the r×r matrix whose rows are the first r (linearly independent)
charge vectors q(1), q(2), . . . , q(r). With respect to the redefined U(1), we can assemble the
new charge vectors into an Ns × r matrix,
Q′ =
δij
−−−−q′ (k)j
, (14)
where i = 1, 2, . . . , r and k = r + 1, r + 2, . . . , Ns label the Ns rows of the matrix and
j = 1, 2, . . . , r.
One can now write out the spurion contributions to the scalar potential. Using eq. (13),
the generalization of eq. (12) is immediate,
VX′
1,X′
2,...,X′
r=
r∑
i=1
Vi(vn) cos θ′i +
Ns∑
k=r+1
Vi(vn) cos
(r∑
j=1
q′ (k)j θ′j
), (15)
where θ′j is defined in eq. (13). The phases θ′j that explicitly appear in the second term
of eq. (15) are potential CP-violating phases. Generically we would expect r CP-violating
phases when Ns > r. However, if there are r′ columns of zeros below the dashed line in
eq. (14), i.e. for all k = r + 1, . . . , Ns,
q′ (k)j = 0 for r′ values of the index j, (16)
then only r− r′ phases appear in the second term of eq. (15). The r′ phases that are absent
do not acquire nontrivial CP-violating expectation values since for these phases, the analysis
reduces to the first case of Ns = r treated above.
There is a simple basis-independent interpretation of r′. Namely, r′ is equal to the number
of charge vectors that are linearly independent of the remaining Ns−1 charge vectors. Thus,
we conclude the following:
9
For a scalar potential with Ns spurion terms that exhibits SCPV, the number of potential
CP-odd phases is given by d = r − r′. That is d is equal to the difference of the dimension
of the vector space spanned by the Ns charge vectors and the number of charge vectors that
are linearly independent of the remaining Ns − 1 charge vectors.
Note that the above result automatically incorporates the case of Ns = r treated above,
where all the charge vectors are linearly independent, in which case r′ = r and d = 0. That
is, there is no SCPV when Ns = r as expected. A geometrical interpretation of the result
d = r − r′ is given in Appendix A.
As a simple example (which corresponds to the chiral Lagrangian of Section V.A), consider
the charge vectors (1, 0) , (0, 1) , (−1,−1). In this example, Ns = 3 and N = r = 2.
However, note that none of the charge vectors is linearly independent of the other two
charge vectors. In each case, we can express a given charge vector as a linear combination
of the other two. Hence, in this example, r′ = 0 and we conclude that d = r− r′ = 2. Thus,
in this example there are two potential CP-violating phases that characterize the vacuum.
If at least one of the r− r′ remaining nontrivial phases differs from a multiple of π at the
minimum of VX′
1,X′
2,...,X′
r[cf. eq. (15)], then the model exhibits SCPV. Generically, such a
solution will exist if the scalar potential parameters satisfy certain conditions. In particular,
there will be a continuous range of scalar potential parameters that yields a continuous
range of values for the CP-violating phase(s). Although we have implicitly assumed that
the coefficients of each spurion contribution to the scalar potential are independent, our
analysis also applies to cases in which the coefficients of inequivalent spurions are related
due to, e.g., a discrete symmetry of the scalar potential. In some scenarios of this kind,
SCPV occurs independently of the choice of the remaining free scalar potential parameters
(after the discrete symmetry is imposed), in which case the corresponding CP-violating
phases may take on only nontrivial discrete values. An example of such a phenomenon is
the so-called geometrical CP-violation of [23].
IV. EXAMPLE: SCPV IN THE TWO HIGGS DOUBLET MODEL
The two Higgs doublet model (2HDM) provides a good theoretical laboratory for ap-
plying the results of the previous section. Some of the results in this section are known.
Nevertheless, we reproduce them here in a very simple and clear fashion by using our the
group-theoretic approach established in sec. III.
The 2HDM consists of two hypercharge-one, SU(2)L doublets (Φ1,Φ2). The SU(2)L×U(1)Y
gauge-covariant kinetic energy terms possess an SU(2)L×U(1)Y×SU(2)F symmetry, where
the SU(2)F corresponds to a “Higgs-flavor” symmetry transformation, Φi → UijΦj with
10
U ∈ SU(2)F . The generic 2HDM potential,
V = m211Φ
†1Φ1 +m2
22Φ†2Φ2 −
(m2
12Φ†1Φ2 + h.c.
)
+ 12λ1
(Φ†
1Φ1
)2+ 1
2λ2
(Φ†
2Φ2
)2+ λ3Φ
†1Φ1Φ
†2Φ2 + λ4Φ
†1Φ2Φ
†2Φ1
+
[12λ5
(Φ†
1Φ2
)2+ λ6Φ
†1Φ1Φ
†1Φ2 + λ7Φ
†2Φ2Φ
†1Φ2 + h.c.
], (17)
breaks the SU(2)F Higgs flavor symmetry completely.
Since there are four complex degrees of freedom, there are four potentially physical SCPV
phases, related to the four diagonal generators
1ij1αβ, 1ijT3αβ , T 3
ij1αβ, T 3ijT
3αβ , (18)
acting on the Φiα, where SU(2)F (L) indices are denoted by Roman (Greek) indices. The first
two are the diagonal generators of the SU(2)L×U(1)Y gauge symmetry, and thus cannot give
rise to SCPV, as discussed in section III. As for T 3ij1αβ, which generates the Peccei-Quinn
(PQ) symmetry [10, 11] (Φ1 → eiαΦ1 and Φ2 → e−iαΦ2), it is not gauged and is generically
broken by the scalar potential. Therefore it can potentially trigger SCPV. The last generator
T 3ijT
3αβ (“chiral PQ”) cannot give rise to SCPV in those vacua that preserve electric charge.
In particular the two VEVs are aligned in the U(1)EM preserving vacuum, in which case
chiral PQ becomes degenerate with PQ.
In order to find models with SCPV, we choose a basis in which all the parameters in
eq. (17) are real. In this basis, we must then explicitly break the U(1)PQ. We now perform
a U(1)PQ spurion analysis.4 The various parameters transform formally under U(1)PQ as
follows. The parameters m211, m
222, and λ1,2,3,4 are neutral with respect to U(1)PQ, whereas
the other parameters possess PQ charges:
m212[2], λ5[4], λ6[2], λ7[2], (19)
where we have assigned the fields with Φ1[1], Φ2[−1].
In light of the above charge assignment, SCPV can arise in a realistic setting only if
1. λ5 is turned on.
2. At least one of the couplings m212, λ6, or λ7 is turned on.
3. The other 2HDM parameters are chosen such that the SU(2)L×U(1)Y gauge symmetry
is broken to U(1)EM.
4 A more general SU(2)F spurion analysis is also quite useful for other 2HDM applications. See Appendix B
for further details.
11
Consider the following simple example (the general case is treated in Appendix B of [24]):
m211, m
222 < 0, m2
12 = 0,
λ1,2 > 0, λ5,6 6= 0, λ3 = λ4 = λ7 = 0, (20)
where |λ5,6| ≪ λ1,2. In this case,
〈Φi〉 ≃
0√m2
ii/λi
, (21)
with small corrections of order O (λ5,6/λ1,2). We see that U(1)PQ is broken only by the terms
VPQ = 1
2λ5
(Φ†
1Φ2
)2+ λ6Φ
†1Φ1Φ
†1Φ2 + h.c. (22)
We parametrize the two expectation values as
Φ01 = v1e
iθeiϕ, Φ02 = v2e
iθe−iϕ. (23)
The new terms induce a potential for the otherwise flat ϕ, which is given by
∆V = λ5v21v
22 cos(4ϕ) + 2λ6v
31v2 cos(2ϕ). (24)
For parameters in the range |λ6| tanβ < 2λ5, this potential is minimized at
cos(2ϕmin) =λ6
2λ5tan β, (25)
where tanβ ≡ v1/v2, resulting in spontaneous CP violation.
V. OTHER MODELS OF SPONTANEOUS CP VIOLATION
In this section, we briefly examine other models that exhibit SCPV, in light of the nec-
essary conditions developed in Section III.
A. The Chiral Lagrangian
Dashen’s model of spontaneous CP violation [12] is based on the three-flavor chiral La-
grangian (see e.g. [25] for a modern review). Recall that this theory is the low energy
description of three-flavor QCD, and it describes the spontaneous breaking of
SU(3)L × SU(3)R → SU(3)V , (26)
where the two SU(3) groups act on the left- and right-handed quarks (u, d, s), respectively.
The vacuum transforms as (3, 3) under SU(3)L × SU(3)R:
Σ0 → L(εaL) Σ0R†(εbR) . (27)
12
In order to ensure that only the diagonal SU(3)V transformations (L = R) leave the vacuum
invariant as required by eq. (26), it follows that Σ0 = 1.
Note that the condition Σ0 = 1 is basis dependent. Indeed, one can simply redefine all
(3, 3) fields by applying an arbitrary SU(3)L × SU(3)R transformation. As a result of such
a field redefinition,
Σ0 = U , U ∈ SU(3) . (28)
Relative to the new basis, the symmetry-breaking pattern is SU(3)L × SU(3)R → SU(3)U ,
where an SU(3)U transformation corresponds to R = U †LU in eq. (27).
As a consequence of the spontaneous breaking of chiral symmetry, there are eight Gold-
stone modes Ga = πi, Ki, η, which are parameterized as
G(x) ≡ Ga(x)T a =1√2
1√2π0 + 1√
6η π+ K+
π− − 1√2π0 + 1√
6η K0
K− K0 − 2√6η
, (29)
where the T a are the SU(3) generators in the fundamental representation. The chiral La-
grangian is expressed in terms of the (3, 3) field Σ(x), which depends on the Goldstone fields
via
Σ(G) = eiG(x)/fΣ0eiG(x)/f , (30)
where Σ0 ≡ 〈Σ〉. In the case of Σ0 = 1, the Goldstone fields transform linearly under the
vector SU(3)V and transform nonlinearly and non-homogeneously under the spontaneously
broken axial transformations, for which L = R†. The non-homogeneous term of the trans-
formation law is a signal that the Goldstone fields are massless and derivatively coupled, as
long as there are no explicit SU(3)L×SU(3)R breaking terms in the Lagrangian. Of course,
these conclusions do not depend on the choice of Σ0 = 1, since all vacua related by the
matrix U in eq. (28) are equivalent.
However, in order for the chiral Lagrangian to describe nature, the chiral symmetry must
be broken explicitly. Such explicit breaking is introduced both by electromagnetic gauge
interactions and by the quark masses. The chiral Lagrangian takes the form
L = 14f 2Tr
(DµΣ
†DµΣ)+ 1
2B0f
2Tr(MΣ† + ΣM †) , (31)
where B0 is proportional to the quark-antiquark condensate (see, e.g., [26]),
M =
mu 0 0
0 md 0
0 0 ms
, (32)
and Dµ is the gauge covariant derivative. Once explicit chiral symmetry-breaking is intro-
duced, all vacua related by the matrix U in eq. (28) are no longer equivalent. In particular,
13
vacua corresponding to different eigenvalues of U are now inequivalent (e.g. they have dif-
ferent energy values). For example, if the quark masses are all positive, then the potential
energy due to the explicit chiral symmetry breaking is minimized by assuming that Σ0 = 1.
However, it is possible that some of the quark mass parameters are negative.5 Without loss
of generality, one can choose the vacuum value Σ0 = U to be diagonal. Since U ∈ SU(3),
the diagonal elements are pure phases whose product is equal to one. That is,
Σ0 =
eiθu 0 0
0 eiθd 0
0 0 e−i(θu+θd)
. (33)
Dashen’s observation was that a region exists in the (mu, md, ms) parameter space where θuand θd are not minimized at the origin, thus inducing SCPV. The potential for the phases is
V = B0f2 [mu cos θu +md cos θd +ms cos (θu + θd)] . (34)
Provided that mumd < 0,6 the potential above is minimized when [25]
mu sin θu = md sin θd = −ms sin (θu + θd) . (35)
It is convenient to introduce dimensionless mass ratios,
x ≡ mu/ms , y ≡ md/ms . (36)
Assuming xy < 0, we can use eq. (35) to obtain the vacuum values of θu and θd,
cos θu = 12
(y
x2− 1
y− y
), cos θd =
12
(x
y2− 1
x− x
), (37)
under the assumption that −1 ≤ cos θu,d ≤ 1. If this latter assumption is false, then
the minimum of the potential for the phases lies on the boundary where | cos θu,d| = 1,
corresponding to a CP-conserving vacuum. Thus, SCPV can arise if and only if xy < 0 and
−1 < cos θu,d < 1. Using eq. (37), these inequalities yield 7
|x|1 + |x| < |y| < |x|
1− |x| , xy < 0 , (38)
in which case the vacuum is characterized by two independent physical phases θu and θdgiven by eq. (37). In Fig. 1, we show regions of the x–y plane that admit SCPV. Indeed,
this range is ruled out phenomenologically (using the light quark masses quoted in [27]).
5 The physical quark masses are given by the absolute values of the quark mass parameters. Nevertheless,
the signs of the quark masses can have physical relevance, as the present discussion makes clear.6 For mumd > 0, the extremum condition given by eq. (35) is a local maximum.7 Note that if we interchange x and y in eq. (38), the results are identical to the original inequalities.
14
FIG. 1. Regions of the parameter space of Dashen’s Model parameter space of Dashen’s model,
where spontaneous CP violation occurs [cf. eq. (38)]. A point in this parameter space corresponds
to (x, y) ≡ (mu/ms,md/ms). The size of the phase θu is shown, with maximum values depicted
in dark (blue), and minimum values in light (yellow). In these regions, θd also acquires a nonzero
value, as explained in the text. The value of θd at the point (x, y) is equal to the value of θu at the
point (y, x).
Although Dashen’s model is no longer a viable model for CP-violation, we can use this
model to illustrate the results of Section III.B in the case of more than one U(1) factor.
Prior to turning on the explicit breaking terms (namely the spurions mu, md and ms), there
are two spontaneously-broken U(1) generators that can be identified with the two diagonal
SU(3) generators T 3 and T 8. In fact, it is more convenient to define linear combinations of
these two generators,
Tu ≡ T 3 +√3 T 8 =
1 0 0
0 0 0
0 0 −1
, Td ≡ −T 3 +
√3 T 8 =
0 0 0
0 1 0
0 0 −1
, (39)
which can be used to shift the values of θu and θd, respectively. Applying Tu and Td to
the vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) yields the U(1)u and U(1)d charges of the three
spurions, respectively. The corresponding charge vectors are given by:
mu(1, 0), md(0, 1), ms(−1,−1). (40)
15
The three charge vectors are linearly dependent and span a two-dimensional vector space.
In the notation of III, we have Ns = 3 > rkQ = 2, in which case SCPV is possible. Indeed
the conditions for SCPV derived in Section III.B, when applied to the above set of spurions,
yields potentially two independent physical CP-violating phases θu, θd that characterize the
vacuum.
Had we considered a chiral Lagrangian based on U(3)L×U(3)R instead of SU(3)L×SU(3)R,
then Σ0 = diag(eiθu , eiθd , eiθs), with no relation among the three phases. Prior to turning
on the explicit breaking terms, there are now three spontaneously-broken U(1) generators
that can be identified with Tu, Td and T 0, where T 0 is the 3 × 3 identity matrix which
generates an axial U(1)A transformation. The corresponding charge vectors of the spurions,
mu(1, 0, 1), md(0, 1, 1), ms(−1,−1, 1) , (41)
are linearly independent, spanning the full three-dimensional vector space, so that Ns=rk Q.
Naively, it seems that none of the three phases is physical, resulting in the absence of SCPV.
However, the axial U(1)A symmetry is anomalous, and can be modeled by adding an explicit
U(1)A breaking term to the chiral Lagrangian that is proportional to (ln det Σ)2 [28–30].
Consequently, there is a fourth spurion so that Ns = 4 > rkQ = 3, and we again conclude
that SCPV is possible. The corresponding fourth charge vector is (0, 0, 1); hence the analysis
of Section III implies that there are three potential physical CP-violating phases θu, θd and θsthat characterize the vacuum. Hence, including the axial U(1)A symmetry and its anomaly-
induced explicit breaking does not spoil the existence of a SCPV phase in the parameter
space of the chiral Lagrangian. A more detailed study is presented in [30], where the effect
of the strong CP angle θ is also taken into account.
Finally, it is noteworthy that in the case of two light flavors, the effective potential of the
SU(2)L×SU(2)R theory depends only on cos θu = cos θd, whereas the corresponding spurions
are mu(1) and md(−1). Using the nomenclature of Section III, there is only one inequivalent
spurion, in which case the SU(2)L×SU(2)R chiral Lagrangian cannot give rise to SCPV.
B. The Minimal Nelson-Barr Model
Here we will consider the model by Bento, Branco, and Parada [13]. The model solves the
strong CP problem by imposing CP as an exact symmetry, and breaking it spontaneously,
thereby producing unsuppressed CKM phase, along with a suppressed strong CP phase.
The field content of the model is the SM plus one gauge singlet complex scalar S, and
one pair of vector-like down quarks DL, DR. The new interactions of the Lagrangian are
given by
δL = −µDLDR − (fiS + f ′iS
∗)DLdiR + h.c. (42)
Moreover, due to the presence of terms such as S2, S4, etc., there is a range of parameter
space for which 〈S〉 = V eiα. The phase α eventually feeds into the SM fermion mass
16
matrices and provides the sole source of CP violation. Since the couplings fi and f ′i are
flavor-dependent, this phase can become the CKM phase once both scalars acquire VEVs.
The radiatively induced strong CP-violating parameter θ is small and therefore the strong
CP problem is solved.
In terms of our group theoretical condition, the scalar sector has a single spontaneously
broken U(1) that is explicitly broken by more than one spurion (e.g., S2 and S4), such that
the vacuum has one physical nonzero phase. This is similar to the toy model with one
complex scalar field presented in section III.
C. Little Higgs models
The Little Higgs framework is a class of nonlinear sigma models that produce the SM
as their low energy limit. By careful design, dubbed “collective symmetry breaking” [31],
the Higgs mass parameter does not receive quadratically divergent corrections at one-loop.
These models can potentially solve the little hierarchy problem, since it allows for the Higgs
mass to be of O(100 GeV) even when the UV cutoff is as high as O(10 TeV).
A popular little Higgs model is the Littlest Higgs model of [32], in which SU(5) is broken
to SO(5) by a two index symmetric SU(5) tensor. The Lagrangian is given by
L =f 2
8Tr |DµΣ|2 + λ1fQiΩ
itR + λ′f t′Lt′R + h.c., (43)
where f is the Goldstone decay constant, Qi and t′L,R are fermions, and Ωi is an SU(5)
breaking function of Σ elements that is chosen in accordance with the principle of collective
symmetry breaking.
In a variant of this model [33], there is an exact global U(1) which is spontaneously
broken [14]. This U(1) is generated by Y ′ = diag(1, 1,−4, 1, 1). As a result, there is an exact
Goldstone mode η associated with Y ′. In order to make the theory viable, the field η must
acquire mass, requiring explicit breaking of U(1)Y ′. A possible spurion that breaks this U(1)
would be s = (0, 0, 1, 0, 0)T , transforming (formally) under the fundamental representation
of SU(5). Its symmetry breaking pattern is SU(5) → SU(4), which acts on the (3, 3) minor.
The nine broken generators include Y ′ and generators which are also broken by the gauging.
In particular, any function of Σ33 = s†Σs would break Y ′ while maintaining gauge invariance.
The term,
δL = εf 4Σ33 + h.c., (44)
is sufficient to generate mass for the Goldstone boson η [cf. eq. (31)]. However, a physical
CP-odd phase can arise only in the presence of at least two different terms. As a simple
example, consider
δLSCPV = εf 4(aΣ33 + bΣ2
33
)+ h.c., (45)
17
where we take ε, a, b to be real, with a, b ∼ O(1) and ε loop-suppressed. This results in the
following tree-level potential for η:
Vη = 2εf 4
(a cos
2η√5f
+ b cos4η√5f
). (46)
This potential is minimized for
〈η〉 = 12
√5f arccos
(−a
4b
)if
∣∣∣a
4b
∣∣∣ < 1, (47)
which is of order one if we assume no hierarchy between a and b.
Further discussion of CP violation in this class of models and related issues can be found
in [14].
VI. CONCLUSIONS AND FUTURE DIRECTIONS
We have formulated the necessary conditions for spontaneous CP violation from a group-
theoretic perspective, i.e., in terms of breaking patterns of global U(1) symmetry generators.
This new framework allows for a more systematic study of spontaneous CP violation model
building. We have used the fact that CP-violating phases in the vacuum are related to
operators that explicitly break the corresponding U(1) groups and the corresponding spu-
rions which are the coefficients of these operators. Such phases are nontrivial and signal
spontaneous CP-violation only in cases where there are a sufficient number of inequivalent
spurions relative to the number of broken U(1) generators.
We assume that the scalar potential of the model is explicitly CP-conserving. In the case
of a single CP-violating phase that characterizes the vacuum, the phase is physical only if
the associated U(1) is broken explicitly by at least two spurions whose U(1) charges differ in
magnitude. We have generalized this result to the case of multiple phases and the associated
U(1) factors. To each spurion, one can assign a charge vector whose components are the
U(1) charges. Two spurions are called equivalent if their charge vectors are equal (up to a
possible overall minus sign). If there are Ns inequivalent spurions whose charge vectors span
an r-dimensional vector subspace, then there is at least one potential physical CP-violating
phase that characterizes the vacuum only if Ns > r. The number of potential CP-odd
phases is then determined to be equal to r − r′, where r′ is the number of charge vectors
that are linearly independent of the remaining Ns−1 charge vectors. The actual value of the
potential CP-violating phase is ultimately determined by minimizing an effective potential.
If a minimum exists such that at least one CP-violating phase is θCP 6= 0, π, then the CP
symmetry is spontaneously broken.
Using these results, we have analyzed the two Higgs doublet model, Dashen’s model for
spontaneous CP violation in the chiral Lagrangian, a minimal Nelson-Barr model, and the
Littlest Higgs with spontaneous CP violation. For the two-Higgs doublet model, we have
18
also performed a comprehensive spurion analysis, in which we employ the full SU(2) Higgs
flavor group. We reproduce results previously obtained by Ivanov [15], and demonstrate how
to use this formalism to construct invariant relations that are independent of the choice of
scalar field basis.
The applications presented in this paper focus on tree-level results. It is of interest to
consider whether our framework allows for spontaneous CP-violation to be generated by
radiative effects. Consider the case of a single CP-violating phase that characterizes the
vacuum. For this to be a robust result that holds over an O(1) fraction of the model
parameter space, one requires the two inequivalent spurions to be of comparable size. If
one of the spurions arises from a tree-level operator and the other arises radiatively, then it
appears that the latter requirement cannot be satisfied (without violating perturbativity of
the loop expansion).
Nevertheless, one can imagine a number of scenarios in which spontaneous CP-violation
is radiatively generated. For example, in a model with multiple complex scalars, it may
be possible to radiatively generate two inequivalent spurions at the loop level, which could
result in an O(1) CP-violating phase. Alternatively, the tree-level spurions might arise from
a different sector of the theory (such as the fermion sector), in which case one could balance
that against a radiatively generated spurion in the scalar sector. However, the Georgi-Pais
theorem [34] limits the ways in which CP violation can be induced radiatively, without
introducing unnaturally light scalars.
Finally, we note that some of the the global U(1) symmetries related to the CP-violating
phases may be anomalous. In this case, the anomaly is manifested by the presence of
explicitly breaking terms in the Lagrangian. If the terms generated by the anomaly satisfy
the necessary conditions developed in this paper, then one could imagine the possibility of
spontaneous CP-violation whose presence is due to the anomaly. It would be instructive to
find explicit models that realize this possibility. We leave these interesting possibilities for
a future study.
ACKNOWLEDGMENTS
We appreciate useful discussions with Gustavo Branco, Benjamin Grinstein, Richard
Hill, Aneesh Manohar, Patipan Uttayarat and Ivo de Medeiros Varzielas. We are especially
grateful to an anonymous referee who encouraged us to provide a more transparent derivation
of the number of CP-violating phases established in Section III.B. The final stages of this
work was supported in part by the National Science Foundation under Grant No. PHY-
1055293 and the hospitality of the Aspen Center for Physics. H.E.H. is supported in part
by U.S. Department of Energy grant number DE-FG02-04ER41268.
19
Appendix A: Geometrical interpretation of the number of CP-violating phases
We can employ the following geometrical construction for establishing the number of
potential CP-violating phases in a SCPV scalar potential. Using the notation of Section
III.B, we first select a linearly independent set of r charge vectors and use this set as a
basis for the linear space of charge vectors. This basis can be used to construct hyperplanes
spanned by a subset of the basis vectors. For example, each basis vector defines a one-
dimensional line that lies parallel to the corresponding basis vector, each pair of basis vectors
spans a two-dimensional plane, etc. Consider the set of all such hyperplanes. From this set,
we can identify the unique hyperplane of minimal dimension d that contains the span of the
remaining Ns − r charge vectors. (Note that d must lie in the range 1 ≤ d ≤ r ≤ N .) Each
basis vector that lies in the hyperplane of minimal dimension is associated with a physical
phase. For example, if the remaining charge vectors are all parallel to a single basis vector
then d = 1, in which case there is one potential physical phase. We conclude the following:
The number of potential CP-violating phases is equal to d, obtained by determining the
unique hyperplane of minimal dimension d, constructed from all possible subsets of the r
basis vectors, in which the span of the remaining Ns − r charge vectors resides.
The procedure presented above is inherently geometrical. In particular, the number d does
not depend on the initial choice of the r linearly-independent basis vectors. Hence, the num-
ber of potential CP-violating phases that characterizes the vacuum is a basis-independent
concept. Indeed, it is straightforward to show that this procedure yields the result obtained
in Section III.B. In particular, it is convenient to employ the basis of U(1) generators that
yields the matrix Q′ given in eq. (14). Let r′ be the number of columns of zeros that lie
below the dashed line in eq. (14). Focusing on the remaining r − r′ columns, consider the
row vectors whose 1 appears in one of these r − r′ columns. The span of these row vec-
tors is a hyperplane of dimension d = r − r′, which we identify as the number of potential
CP-violating phases.
As a simple example, we again consider the charge vectors (1, 0) , (0, 1) , (−1,−1),where Ns = 3 and N = r = 2. For any c 6= 0, the vector c(−1,−1) is neither parallel to
(1, 0) nor to (0, 1). Indeed c(−1,−1) lies in the two dimensional plane spanned by (1, 0)
and (0, 1), so that hyperplane of minimal dimension that contains c(−1,−1) is a plane
of dimension d = 2. Thus, in this case there are two potential CP-violating phases that
characterize the vacuum.
Note that the above analysis applies trivially to the case of a single complex scalar field,
where N = r = d = 1. In this case, there is one potential CP-violating phase if Ns > r,
which yields Ns > 1. This conclusion coincides with the analysis given in Section III.A for
the case of one complex scalar field.
20
Appendix B: The 2HDM spurion analysis and some applications
1. Full SU(2)F spurion analysis
We begin by expressing all the parameters in the 2HDM scalar potential in terms of
invariants and spurions of SU(2)F . This is accomplished by constructing gauge invariant
terms from the fields Φiα, Φiα ≡ (Φiα)
† and the invariants ǫij , ǫαβ , δβα, δji [15, 35–38]. The
only gauge-invariant bilinear term is (M2) ij ΦiαΦ
jα, where M2 transforms under SU(2)F as
two-index tensor,
M2 =
m2
11 −m212
−m212 m2
22
, (B1)
with (M2)12 ≡ −m2
12 and m212 ≡ (m2
12)∗. For the gauge-invariant quadrilinear terms, we
start with ΦiαΦjβΦkγΦ
ℓδ, and note that there are two ways to contract all the indices in a
gauge invariant manner. The first invariant is,
ΦiαΦjβΦkγΦ
ℓδǫαβǫγδ Aij
kℓ = ΦiαΦjβΦkγΦ
ℓδ(δαγ δ
βδ − δαδ δ
βγ
)ǫijǫkℓ A , (B2)
where
Aijkℓ = −Aji
kℓ = −Aijℓk = Aji
ℓk , (B3)
is antisymmetric with respect to the separate interchange of upper and lower indices. The
antisymmetry property of Aijkℓ implies that only one independent element exists, Aij
kℓ =
ǫijǫkℓA, where
A = 18(λ3 − λ4) , (B4)
which is a scalar with respect to SU(2)F transformations.
The second invariant is,
ΦiαΦjβΦkγΦ
ℓδ(δαγ δ
βδ + δαδ δ
βγ
)Σij
kℓ (B5)
where
Σijkℓ = Σji
kℓ = Σijℓk = Σji
ℓk , (B6)
is symmetric with respect to the separate interchange of upper and lower indices. In addition,
hermiticity implies that Σijkℓ = Σkℓ
ij . In terms of the parameters in eq. (17), we have
Σ1111 =
14λ1, Σ22
22 =14λ2, Σ12
12 =18(λ3 + λ4),
Σ2211 = Σ11
22 =14λ5, Σ12
11 = Σ1112 =
14λ6, Σ22
12 = Σ1222 =
14λ7. (B7)
Since all the spurions transform as integer spin, they can be expressed as SO(3) ten-
sors [15], labeled by adjoint SU(2)F indices a, b, . . .. The squared-mass term decomposes as
2⊗ 2 = 1⊕ 3. Explicitly,
M2 = 2m2aTa + µ2
1 , (B8)
21
where Ta ≡ 12σa are the SU(2) generators, with normalization Tr(TaTb) =
12δab, and 1 is the
2 × 2 identity matrix. In particular, the antisymmetric part of the tensor product is the
singlet that is given by the trace,
µ2 ≡ 12Tr(M2) = 1
2
(m2
11 +m222
). (B9)
The symmetric part of the tensor product, denoted by (2⊗ 2)sym, is the triplet given by
m2a = Tr
(M2Ta
)=(−Rem2
12 , Imm212 , 1
2(m2
11 −m222)). (B10)
The quadrilinear terms transform as (2⊗ 2)sym ⊗ (2⊗ 2)sym = 1⊕ 3⊕ 5. Explicitly,
Σijkℓ =
12Dab(Ta)k
i(Tb)ℓj + 1
8Pa
[(Ta)k
iδjℓ + (Ta)kjδiℓ + (Ta)ℓ
jδik + (Ta)ℓiδjk]+ 1
24S(δikδ
jℓ + δiℓδ
jk) ,
(B11)
where Dab is a traceless symmetric second-rank tensor. Using the Fierz identity,
(Ta)ki(Tb)ℓ
j = 12
[(Ta)k
j(Tb)ℓi + (Tb)k
j(Ta)ℓi − δab(Tc)k
j(Tc)ℓi + 1
4δabδ
jkδ
iℓ
]
+14iǫabc
[δiℓ(Tc)k
j − δjk(Tc)ℓi], (B12)
it follows that Dab(Ta)ki(Tb)ℓ
j = Dab(Ta)kj(Tb)ℓ
i. Hence, Σijkℓ given by eq. (B11) is symmetric
under the separate interchange of its lower and its upper indices, as required.
Using eq. (B11), the singlet is given by the trace,
S = 4Σijij = λ1 + λ2 + λ3 + λ4, (B13)
the triplet is given by
Pa = 4Σijkℓ (Ta)j
ℓδki = (Re (λ6 + λ7) , −Im (λ6 + λ7) ,12(λ1 − λ2)), (B14)
and the 5-plet is a traceless symmetric second-rank tensor given by
Dab = 2[4Σij
kℓ (Ta)jℓ (Tb)i
k − 13Σij
ijδab
]=
−13∆+Reλ5 −Im λ5 Re (λ6 − λ7)
−Im λ5 −13∆− Reλ5 −Im (λ6−λ7)
Re (λ6−λ7) −Im (λ6−λ7)23∆
,
(B15)
where ∆ ≡ 12(λ1 + λ2)− λ3 − λ4.
The above results are equivalent to the group-theoretical decomposition of the 2HDM
scalar potential obtained in [15].8
8 To obtain Ivanov’s results [15], one simply replaces M2 and Σijkℓ with their complex conjugates in the
above expressions for m2
a, Pa and Dab. Ivanov also introduces different overall normalizations for these
quantities, which are not critical to the applications presented in this Appendix.
22
2. Invariant relations
Having obtained all the SU(2)F spurions, we can now find invariant relations among
parameters, i.e., relations that hold in every basis, provided they hold in one basis. At
the linear level, invariant relations can be obtained either by setting a singlet quantity to
a constant (any constants will do), or by setting the non-singlet spurions to zero. This
procedure yields six invariant linear relations:
1. m211 +m2
22 = 2µ20,
2. λ3 − λ4 = 8A0,
3. λ1 + λ2 + λ3 + λ4 = S0,
4. m211 −m2
22 = m212 = 0,
5. λ1 − λ2 = λ6 + λ7 = 0,
6. 12(λ1 + λ2)− λ3 − λ4,= λ5 = λ6 − λ7 = 0,
where µ20, A0, and S0 are arbitrary constants. This generalizes [5], in which relations 4 and 6
were noted and discussed. If the scalar potential is SU(2)F -invariant, then the relations 4,
5 and 6 above must be simultaneously satisfied, i.e.,
m211 = m2
22 , m212 = 0 , λ1 = λ2 = λ3 + λ4 , λ5 = λ6 = λ7 = 0. (B16)
This particular model was introduced previously in [20] and exhibits the largest allowed
Higgs family symmetry of the 2HDM scalar potential.
Higher order invariant relations may be constructed by forming scalar combinations of
products of the nontrivial spurions m2a, Pa, and Dab. For example, we can obtain invariant
quadratic relations by constructing scalar quantities from the product of two spurions and
setting the result to a constant. For example,
m2am
2a =
∣∣m212
∣∣2 + 14
(m2
11 −m222
)2= const. (B17)
PaPa = |λ6 + λ7|2 + 14(λ1 − λ2)
2 = const. (B18)
m2aPa = Re
[m2
12 (λ6 + λ7)]+ 1
4
(m2
11 −m222
)(λ1 − λ2) = const. (B19)
Tr(D2) = 16(λ1 + λ2 − 2λ2 − 2λ4)
2 + 2|λ5|2 + 2|λ6 − λ7|2 = const. (B20)
An example of an invariant cubic relation is m2aPbDab = 0, and so on.
23
3. Condition for the existence of a U(1)PQ symmetry
In order to exemplify the power of SU(2)F spurion analysis, we derive the condition
for existence of a U(1) global symmetry, which in a particular basis for the scalar fields
coincides with the Peccei-Quinn symmetry and the corresponding U(1) generator is T 3ij1αβ
[cf. eq. (18)]. In an arbitrary basis, the generator of U(1)PQ, denoted by TPQ, must be a
linear combination of the SU(2)F generators. Hence,
TPQ = qaTa =12qaσa , (B21)
which defines the three-vector qa, transforming under the adjoint representation of SU(2)F .
It is convenient to normalize qa such that its squared-length is qaqa = 1. If the scalar
potential preserves the U(1)PQ symmetry, then all spurions must be fixed (up to an overall
scale) by qa. In particular,
m2a = c1qa , Pa = c2qa , Dab = c3
(qaqb − 1
3δab), (B22)
where the ci are arbitrary constants. Eq. (B22) provides an elegant basis-independent set
of conditions for the existence of a PQ symmetry in the 2HDM. One can use the explicit
expressions for m2a, Pa and Dab [cf. eqs. (B10), (B14) and (B15), respectively] to rewrite
eq. (B22) in terms of the 2HDM scalar potential parameters in an arbitrary basis. The
resulting equations are not particularly illuminating, so we do not write them out here.
To verify the above assertion, consider the spurion, M2 = m2aσa + µ2
1 introduced in
eq. (B8). The triplet spurion m2a transforms under the adjoint representation of SU(2)F ,
and therefore breaks the global SU(2)F symmetry down to U(1)PQ [39]. The condition that
the U(1)PQ is preserved is equivalent to the requirement that
[TPQ , M2] = 0 . (B23)
Inserting eqs. (B8) and (B21) into the above condition yields
12qam
2b [σa , σb] = iǫabcqam
2bσc = 0 . (B24)
Eq. (B24) implies that qa ∝ m2a, which identifies the U(1)PQ generator. Indeed, all triplet
spurions must be proportional to qa as indicated in eq. (B22), since any two non-parallel
triplet spurions would completely break the SU(2)F global symmetry [39]. Likewise, the
condition that U(1)PQ is conserved by the spurion Σijkℓ is equivalent to the requirement that
Σmjkℓ (TPQ)m
i − Σijnℓ(TPQ)k
n + Σimkℓ (TPQ)m
j − Σijkn(TPQ)ℓ
n = 0 . (B25)
Using eq. (B11), it follows that
qcDab
(Tb)k
i[Ta , Tc]ℓj + (Tb)ℓ
j [Ta , Tc]ki= iqcDabǫace
[(Tb)k
i(Te)ℓj + (Tb)ℓ
j(Te)ki]= 0 ,
(B26)
which is satisfied by Dab ∝ qaqb − 13δab as indicated in eq. (B22).
24
One is always free to choose a convenient basis for the scalar fields of the 2HDM by
diagonalizing Dab. The eigenvalues of Dab = c3(qaqb − 13δab) are −1
3c3 , −1
3c3 , +
23c3 (note
the doubly-degenerate eigenvalue assuming that c3 6= 0). It is straightforward to check that
Dab is diagonal when qa = (0, 0, 1). Then eq. (B22) implies that m212 = λ5 = λ6 = λ7 = 0
in the D-diagonal basis, which yields the standard form for the 2HDM scalar potential with
PQ-symmetry Φ1 → eiαΦ1 and Φ2 → e−iαΦ2. Moreover, in the D-diagonal basis, we can
identify c1 =12(m2
11 −m222), c2 =
12(λ1 − λ2) and c3 = ∆ = 1
2(λ1 + λ2)− λ3 − λ4.
Of course, eq. (B22) is applicable in an arbitrary basis. These conditions are equivalent
to the invariant conditions given in [4, 20], although the formulation of eq. (B22) is much
simpler and transparent than the conditions originally given. Note that at the exceptional
point of parameter space identified in [4] where m211 = m2
22, m212 = 0, λ1 = λ2 and λ7 = −λ6,
it follows that m2a = Pa = 0. In this case, the condition for PQ symmetry is simply
the existence of a doubly-degenerate eigenvalue of Dab as first noted in [20]. This latter
condition implies that Dab ∝ qaqb − 13δab for some unit vector qa, which then determines the
PQ generator given in eq. (B21).
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